I'm new to soldering, too, plus I have shaky hands which makes it doubly difficult. However, I still managed to do it. Perhaps the most important tip is for you to get TWO soldering irons. Simple pencil irons will work, and they're really cheap. Like $10 or $15. If you don't use two soldering irons at the same time, then I can pretty much guarantee you'll tear off one of the little pads under the capacitor. I lost three motherboards to that problem before it donned on me to use two soldering irons (duh!)
My second suggestion is to get solder paste. It's a little pricey compared to normal solder, but man, I found it to be the best stuff. You just put some of that goop on each of the pads, lay the capacitor right on the goop, then just hold the iron in the goop. All the flux just evaporates, and the solder melts right onto the pads. Super easy, even with shaky hands.
I recommend tantalum capacitors so you can avoid future leakage.
Also, don't push too hard on the caps when you're removing them. When you have both the irons down there melting both sides, the cap will eventually just slide off.